Waveguide-integrated twisted bilayer graphene photodetectors

Graphene photodetectors have exhibited high bandwidth and capability of being integrated with silicon photonics (SiPh), holding promise for future optical communication devices. However, they usually suffer from a low photoresponsivity due to weak optical absorption. In this work, we have implemented SiPh-integrated twisted bilayer graphene (tBLG) detectors and reported a responsivity of 0.65 A W–1 for telecom wavelength 1,550 nm. The high responsivity enables a 3-dB bandwidth of >65 GHz and a high data stream rate of 50 Gbit s–1. Such high responsivity is attributed to the enhanced optical absorption, which is facilitated by van Hove singularities in the band structure of high-mobility tBLG with 4.1o twist angle. The uniform performance of the fabricated photodetector arrays demonstrates a fascinating prospect of large-area tBLG as a material candidate for heterogeneous integration with SiPh.


Supplementary Note 1. The optical conductivity and effective dielectric constant of graphene.
A finite element method mode-solver (COMSOL Multiphysics) was used for the mode analysis of the waveguide-integrated graphene photodetectors (PDs).Graphene is incorporated by an equivalent volume permittivity model.The optical conductivity of single-layer graphene (SLG) was calculated from the Kubo formula, given by: where the σ0 = e 2 /4ℏ is the universal optical conductivity of SLG under the linear-band regime, ω is the incident optical frequency, μ and T are the chemical potential and the temperature of SLG, respectively 1 .In the simulations, the thickness of SLG hG was set to 0.34 nm.The bulk conductivity of SLG is expressed as σ/hG, and the effective dielectric constant of SLG εG(ω) can be described as Equation (S2): () = 1 + () where ε0 is the vacuum permittivity 2 .εG(ω) can be expressed by tensor form: The horizontal component of effective dielectric constant εG(ω) can be obtained by equation (S2), and the vertical component is set as a constant nc = 2.5 3,4 .
Unlike the linear band structure of single layer graphene (SLG) and parabolic-like band structure of AB stacked bilayer graphene (BLG), the Dirac cones of the two individual monolayers in tBLG intersect and form saddle points in reciprocal space, resulting in the formation of van Hove singularities (vHs) in the density of state (DOS) 5,6 .In addition, the position of the intersection (and thus vHs) is θ-dependent.
The energy difference between the vHs can be described as ∆ vHs = 2ℏ F (  2 ).
When the incident photon energy matches ΔEvHs, a pronounced interband transition between vHs and an enhanced absorption happen [3].This can be understood by the joint density of states (JDOS): Where Ec, Ev, k and ℏ are conduction band energy, valance band energy, reciprocal vector, and photon energy, respectively.The integration indicates that the JDOS is highly relevant to the density of states (DOS).For tBLG, when these two singularities match the photon energy, which means the interband transition matches the delt function in the equation R1, the absorption is enhanced.For example, the photon energy of 0.8 eV corresponds to an incident λ of 1,550 nm, θ is thus estimated to be ~4.1°.
The enhanced optical absorption of tBLG can be calculated based on its optical conductivity σtBLG, which comes from two distinct interband transitions 7 : the linearband transition  SLG of SLG and the transition σα from the van Hove singularities (vHs) to the band edges (BEs), as described by σα can be described by the standard Gaussian function: where Sα, Eα, and Γα represent the intensity, energy, and broadening of the optical conductivity peak, respectively.The twist angle-dependent optical conductivity of tBLG was calculated using function (S5), as shown in Fig. 1c.According to the previous theoretical calculation and experiment demonstrations 7,8 , the optical conductivity of tBLG can reach up to 3σSLG when the energy difference of two vHs in tBLG matches the incident wavelength, different from that of AB-stacked BLG (2σSLG).As illustrated in Fig. 1c, σtBLG exhibits a remarkable peak at a twist angle of ~4.1° corresponding to an incident light wavelength of 1,550 nm, and retains greater than 2σSLG, when the θ deviation is within 1°.These calculation results thus indicate that tBLG exhibits larger optical conductivity mainly responsible for enhanced absorption.

Supplementary Note 2. Mode analysis of waveguide-integrated graphene photodetectors.
The absorption enhancement of tBLG is compatible to absorption of field enhancement from plasmonic effect 6 , which means these two enhancement factors can be multiplied to give rise larger enhancement.In principle these two processes are independent: The field enhancement can be seen as "concentration of optical field (or optical beam)" as the light intensity equals the square of the optical electric field".It happens at the optical illuminating process.On the other hand, the absorption enhancement of tBLG happens at the electron excitation process and is due to the enhanced joint density of states (or van Hove singularities).
For a given propagation length of L, the graphene absorptance η(L) can be expressed by ) where ηG is the light absorption ratio of graphene, and α = αG + αM is the total absorption coefficient in dB μm -1 .The light absorption ratio ηG is given by The graphene absorptance is calculated by 9 where P0 is the incident power, l is the coordinate of the line integral along the graphene surface in the xy plane, αe = α/4.34 is the mode absorption coefficient in μm -1 , and AG(l) is the light absorption intensity in W/m 2 of graphene.AG(l) is given by where Et is the transverse component of the electric fields along the graphene surface of the launched waveguide mode.Similarly, the metal absorptance ηM(L) is calculated by Here, the integral area of the xy-plane surface integral is the metal area, and AM (x, y) is the light absorption intensity (W/m 3 ) of metal, given by where εM_im (ω) is the imaginary part of the metal permittivity, and E(x, y) is the electric fields in metal.According to Equation (S8) and (S10), the graphene absorption coefficient αG and the metal absorption coefficient αM are given as (in dB μm -1 ) With Equation (S12) and (S13), the absorption coefficients αG, αM and the graphene absorption ratio ηG can be calculated.The simulation results of the waveguideintegrated graphene PDs operating at 1,550 nm are shown in Supplementary Fig. 3.The optical conductivities and the effective dielectric constant of SLG, AB-stacked BLG, and 4.1° tBLG are given in Supplementary Note 1.As the metal width Wm increases, the metal absorption coefficient αM grows rapidly, while αG saturates when Wm > 200 nm.Therefore, the metal width Wm is set as 200 nm.In this case, (αG, ηG) of SLG, ABstacked bilayer graphene (BLG), and 4.1° tBLG are calculated to be (0.184 dB μm -1 , 52.9%), (0.394 dB μm -1 , 70.8%), and (0.577 dB μm -1 , 78.8%), respectively.The calculated graphene absorptance η(L) varying with the propagation length L is plotted in Fig. 1d in the main text.

Supplementary Note 3. Enhanced optical absorption of 4.1° tBLG at 1,550 nm under surface light illumination.
Previous studies have corroborated the enhanced optical absorption of tBLG covering visible, near-and mid-infrared wavelengths under surface light illumination 6,7,10 .However, it has not been demonstrated for a standard telecom wavelength of 1,550 nm.
To verify this, a photodetector was designed and fabricated on SiO2/Si substrate with a channel simultaneously comprising a SLG domain and a 4.1° tBLG domain.The spatially resolved photocurrent mapping of the device was characterized by scanning photocurrent microscopy.The junction of metal/4.1°tBLG domain generates significantly larger photocurrent (~4 times) than that of the SLG ones at zero bias under 1,550 nm direct illumination (see Supplementary Fig. 6), revealing an enhanced optical absorption of 4.1° tBLG at 1,550 nm.This result also suggests the feasibility of 4.1° tBLG for the construction of waveguide-integrated photodetector demonstrating high photoresponsivity.

Supplementary Note 4. Carrier mobility and contact resistivity of a field-effect transistor (FET) with tBLG as the active material.
Four-probe electrical measurements were used to extract the carrier mobility and the contact resistivity of a tBLG FET.Two probes were used to apply a source-drain voltage VDS and measure the source-drain current IDS, and the other two probes were used to measure the voltage drop along the graphene channel between V1 and V2 (see Fig. 2c in the main text).A global gate was introduced by fabricating the FET on a highly doped silicon substrate capped with a 300-nm thick SiO2 as gate dielectric.Gate voltage VG was swept from 60 to −60 V.The resistance of the whole channel, RDS, and the channel region between probe 1 and 2, R12 can be expressed as follows: The carrier mobility μ can be calculated by: where L12 is the length between probe 1 and 2, W is the channel width, and COX is the gate insulator capacitance per unit area.The contact resistivity RC is given by: where LDS is the channel length.With these formulas, the hole (electron) mobility μ of 10,600 (9,100) cm 2 V -1 s -1 was obtained, and the contact resistivity in this device was estimated to be ~500 Ω cm (see Supplementary Fig. 7).Additionally, given that the contact resistance can be expressed as 2RC/W, where RC is the contact resistivity, and W is tBLG length.Therefore, the contact resistance in our tBLG photodetector is ~125 Ω considering our tBLG length is 8 μm.

Supplementary Note 5. The mechanisms of the waveguide-integrated tBLG photodetectors.
The mechanisms of graphene photodetection include photo-thermoelectric (PTE), photoconductive (PC) and photo-bolometric (PB) effects.PTE effect origins from diffusivity (Seebeck coefficient) differences of hot electrons at different doping levels 11 .It dominates when the source-drain bias is zero or small.In PTE effect, the photocurrent polarity (flowing direction) shouldn't change with source-drain bias polarity.PC effect origins from the migration of hot electrons under source-drain bias, which dominates at relatively large source-drain bias 12,13 .In this regime, the polarity of photocurrent should be the same with that of the source-drain bias.As we have explained in the last response, the PB effect origins from resistivity differences at different lattice temperature 12,13 .In this regime, light illumination heats the lattice, increase the resistivity, and decrease the total current, so the "photocurrent', defined as difference of current with and without illumination, should show opposite polarity with the sourcedrain bias.Therefore, with the polarity of photocurrent as key evidence 11 and the same polarity of photocurrent with source-drain bias in Supplementary Fig. 10, We conclude that the PC effect is the dominating mechanism in our devices.The device performance is almost symmetric under positive and negative bias voltages in Supplementary Fig. 11 because the bias voltages act as photocurrent extracting "forces" and should extract similar scale but opposite polarities.The other notable feature is the nonlinear dependence of photocurrent (responsivity) with bias voltage.This is due to that with voltage increasing the extracting efficiency increases and the main bottle neck lies on the photocurrent numbers which is mainly dependent on incident power 11 .

Table 1 | Comprehensive comparisons of the key figures of merit (FoMs) of waveguide-integrated graphene PDs in literature reports and our results.
S30Supplementary